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Lectures on SchrammLoewner Evolution N. Berestycki & J.R. Norris January 14, 2016 These notes are based on a course given to Masters students in Cambridge. Their scope is the basic theory of SchrammLoewner evolution, together with some underlying and related theory for conformal maps and complex Brownian motion. The structure of the notes is in uenced by our attempt to make the material accessible to students having a working knowledge of basic martingale theory and It o calculus, whilst keeping the prerequisities from complex analysis to a minimum. 1Contents 1 Riemann mapping theorem 4 1.1 Conformal isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 M obius transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Martin boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 SLE(0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Loewner evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Brownian motion and harmonic functions 9 2.1 Conformal invariance of Brownian motion . . . . . . . . . . . . . . . . . . 9 2.2 Kakutani's formula and the circle average property . . . . . . . . . . . . . 11 2.3 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Harmonic measure and the Green function 14 3.1 Harmonic measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 An estimate for harmonic functions (?) . . . . . . . . . . . . . . . . . . . . 15 3.3 Dirichlet heat kernel and the Green function . . . . . . . . . . . . . . . . . 16 4 Compact H-hulls and their mapping-out functions 19 4.1 Extension of conformal maps by re ection . . . . . . . . . . . . . . . . . . 19 4.2 Construction of the mapping-out function . . . . . . . . . . . . . . . . . . 20 4.3 Properties of the mapping-out function . . . . . . . . . . . . . . . . . . . . 21 5 Estimates for the mapping-out function 23 5.1 Boundary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Continuity estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 Di erentiability estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6 Capacity and half-plane capacity 26 6.1 Capacity from1 inH (?) . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.2 Half-plane capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7 Chordal Loewner theory I 29 7.1 Local growth property and Loewner transform . . . . . . . . . . . . . . . . 29 7.2 Loewner's di erential equation . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.3 Understanding the Loewner transform . . . . . . . . . . . . . . . . . . . . 31 8 Chordal Loewner theory II 32 8.1 Inversion of the Loewner transform . . . . . . . . . . . . . . . . . . . . . . 32  8.2 The Loewner ow onR characterizes K \R (?) . . . . . . . . . . . . . . . 34 t 8.3 LoewnerKufarev theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 29 SchrammLoewner evolutions 36 9.1 Schramm's observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 9.2 RohdeSchramm theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 9.3 SLE in a two-pointed domain . . . . . . . . . . . . . . . . . . . . . . . . . 37 10 Bessel ow and hitting probabilities for SLE 39 10.1 Bessel ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 10.2 Hitting probabilities for SLE() on the real line . . . . . . . . . . . . . . . 43 11 Phases of SLE 44 11.1 Simple phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 11.2 Swallowing phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 12 Conformal transformations of Loewner evolutions 47 12.1 Initial domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 12.2 Loewner evolution and isomorphisms of initial domains . . . . . . . . . . . 49 13 SLE(6), locality and percolation 53 13.1 Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 13.2 SLE(6) in a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 13.3 Smirnov's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 14 SLE(8=3) and restriction 58 14.1 Brownian excursion in the upper half-plane . . . . . . . . . . . . . . . . . . 58 14.2 Restriction property of SLE(8=3) . . . . . . . . . . . . . . . . . . . . . . . 59 14.3 Restriction measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 15 SLE(4) and the Gaussian free eld 64 15.1 Conformal invariance of function spaces . . . . . . . . . . . . . . . . . . . . 64 15.2 Gaussian free eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 15.3 Angle martingales for SLE(4) . . . . . . . . . . . . . . . . . . . . . . . . . 70 15.4 SchrammSheeld theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 72 16 Appendix 75 16.1 Beurling's projection theorem . . . . . . . . . . . . . . . . . . . . . . . . . 75 16.2 A symmetry estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 16.3 A Dirichlet space estimate for Brownian motion . . . . . . . . . . . . . . . 78 31 Riemann mapping theorem We review the notion of conformal isomorphism of complex domains and discuss the ques- tion of existence and uniqueness of conformal isomorphisms between proper simply con- nected complex domains. Then we illustrate, by a simple special case, Loewner's idea of encoding the evolution of complex domains using a di erential equation. 1.1 Conformal isomorphisms We shall be concerned with certain sorts of subset of the complex plane C and mappings between them. A set DC is a domain if it is non-empty, open and connected. We say that D is simply connected if every continuous map of the circlefjzj = 1g into D is the restriction of a continuous map of the discfjzj 6 1g into D. A convenient criterion for a domain D C to be simply connected is that its complement in the Riemann sphere Cf1g is connected. A domain is proper if it is not the whole ofC. The open unit disc D =fjzj 1g, the open upper half-plane H =fRe(z) 0g, and the open in nite strip S =f0 Im(z) 1g are all examples of proper simply connected domains. 0 A holomorphic functionf on a domainD is a conformal map iff (z) =6 0 for allz2D. 0 We call a bijective conformal map f :DD a conformal isomorphism. In this case, the 0 1 0 image D =f(D) is also a domain and the inverse map f :D D is also a conformal z map. Every conformal map is locally a conformal isomorphism. The function z7 e is conformal on C but is not a conformal isomorphism on C because it is not injective. We note the following fundamental result. A proof may be found in 1. Theorem 1.1 (Riemann mapping theorem). Let D be a proper simply connected domain. Then there exists a conformal isomorphism  :DD. We shall discuss ways to specify a unique choice of conformal isomorphism  :DD or  : D H in the next two sections. In general, there is no usable formula for  in terms of D. Nevertheless, we shall want to derive certain properties of  from properties of D. We shall see that Brownian motion provides a useful tool for this. 1.2 M obius transformations A M obius transformation is any function f onCf1g of the form az +b f(z) = (1) cz +d where a;b;c;d2 C and adbc =6 0. Here f(d=c) =1 and f(1) = a=c. Mobius  transformations form a group under composition. A M obius transformation f restricts to a conformal automorphism of H if and only if we can write (1) with a;b;c;d2 R and adbc = 1. For 2 0; 2) and w2D, de ne  onD by ;w zw i  (z) =e : ;w 1w z 4Then  is a conformal automorphism of D and is the restriction of a M obius transfor- ;w mation toD. De ne :HD by iz (z) = : i +z Then is a conformal isomorphism and extends to a M obius transformation. The following lemma is a basic result of complex analysis. We shall give a proof in Section 2. Lemma 1.2 (Schwarz lemma). Let f :DD be a holomorphic function with f(0) = 0. i Thenjf(z)j6jzj for all z. Moreover, ifjf(z)j =jzj for some z =6 0, then f(w) =e w for all w, for some 2 0; 2). 1 Corollary 1.3. Let  be a conformal automorphism of D. Set w =  (0) and  = 0 arg (w). Then  =  . In particular  is the restriction of a M obius transformation to ;w  D and extends to a homeomorphism ofD. 1 Proof. Set f =   . Then f is a conformal automorphism of D and f(0) = 0. Pick 0;w u2 Dnf0g and set v = f(u). Note that v 6= 0. Now, eitherjf(u)j =jvj juj or 1 jf (v)j =jujjvj. In any case, by the Schwarz lemma, there exists 2 0; 2) such that i 0 i 2 f(z) = e z for all z, and so  = f  =  . Finally,  (w) = e =(1jwj ) so 0;w ;w ;w =. Corollary 1.4. Let D be a proper simply connected domain and let w2 D. Then there 0 exists a unique conformal isomorphism  :DD such that (w) = 0 and arg (w) = 0. Proof. By the Riemann mapping theorem there exists a conformal isomorphism  :D 0 0 D. Set v =  (w) and  = arg (w) and take  =   . Then  : D D is 0 ;v 0 0 0 a conformal isomorphism with (w) = 0 and arg (w) = 0. If is another such, then 1 0 f =  is a conformal automorphism ofD withf(0) = 0 and argf (0) = 0, sof =  0;0 which is the identity function. Hence  is unique. 1.3 Martin boundary 1 The Martin boundary is a general object of potential theory . We shall however limit our discussion to the case of harmonic functions in a proper simply connected complex domain D. In this case, the Riemann mapping theorem, combined with the conformal invariance of harmonic functions, allows a very simple approach. Make a choice of conformal isomor- 0 0 phism  : D D. We can de ne a metric d on D by d (z;z ) =j(z)(z )j. Then   d is locally equivalent to the original metric but possibly not uniformly so. Say that a  sequence (z : n2 N) in D is D-Cauchy if it is Cauchy for d . Since every conformal n   automorphism of D extends to a homeomorphism of D, this notion does not depend on 2 the choice of . Write D for the completion of D with respect to the metric and de ne the Martin boundary D = DnD. The set D does not depend on the choice of  and 1 See for example 2 2 0 This is the set of equivalence classes of D-Cauchy sequences z = (z : n 2 N), where z  z if n 0 0 (z ;z ;z ;z ;::: ) is also a D-Cauchy sequence. 1 2 1 2 5D ϕ Figure 1: Two distinct points of D and their images under '. nor does its topology. This construction ensures that the map  extends uniquely to a  homeomorphism D D. It follows then that every conformal isomorphism of proper 0 0 simply connected domainsDD has a unique extension as a homeomorphism DD . We abuse notation in writing (z) for the value of this extension at points z2D. Write D for the boundary of D as a subset of C, that is the set of limit points of D in C, which in general is not identi able with D. For b2D, we say that a simply connected subdomain N D is a neighbourhood of b in D iffz2 D :jz(b)j "g (N) for some " 0. A Jordan curve is a continuous injective map :DC. Say D is a Jordan domain if D is the image of a Jordan curve. It can be shown in this case that any conformal   isomorphismDD extends to a homeomorphismDD, so we can identifyD withD. On the other hand, a sequence (z :n2N) inH isH-Cauchy if either it converges inC or n jzj1 asn1. Thus we identifyH withRf1g. For the slit domainD =Hn(0;i n and, for z2 0;i), the sequences (z + (1 +i)=n : n2 N) and (z + (1 +i)=n : n2 N) + are D-Cauchy but are not equivalent, so their equivalence classes z and z are distinct Martin boundary points. Corollary 1.5. Let be a conformal automorphism ofH. If(1) =1, then(z) =z+ for all z2H, for some  0 and 2R. If (1) =1 and (0) = 0, then (z) =z for all z2H, for some  0. 1 Proof. Set  =(0) and  =(1)(0). Since  is a conformal automorphism ofD, we know by Corollary 1.3 that  is a M obius transformation of H, so (z) = (az + b)=(cz +d) for allz2H, for somea;b;c;d2R withadbc = 1. This formula extends by continuity to H =Rf1g. So we must have c = 0,  =b=d and  =a=d 0. Corollary 1.6. LetD be a proper simply connected domain and letb ;b ;b 2D, ordered 1 2 3 anticlockwise. Then there exists a unique conformal isomorphism  : D H such that (b ) = 0, (b ) = 1 and (b ) =1. 1 2 3 6Proof. By the Riemann mapping theorem there exists a conformal isomorphism  :D 0 1 D. Set  = arg (b ) and take  =    . Then  :DH is a conformal 0 3 1 ;0 0 1 isomorphism, and   (b ) =1 so  (b ) =1. Now  (b )  (b ) so there exist ;0 0 3 1 3 1 1 1 2 2 (0;1) and2R such that (b )+ = 0 and (b )+ = 1. Set(z) = (z)+ 1 1 1 2 1 then :DH is a conformal isomorphism satisfying the given constraints. If is another 1 such then f =  is a conformal automorphism of H with f(0) = 0, f(1) = 1 and f(1) =1. Hence f(z) =z for all z2H and so  is unique. Note that in both Corollary 1.4 and Corollary 1.6, we obtain uniqueness of the conformal map by the imposition of three real-valued constraints. 1.4 SLE(0) This section and the next are for orientation and do not form part of the theoretical  development. Consider the (deterministic) process ( ) in the closed upper half-planeH t t0 given by p = 2i t: t This process belongs to the family of processes (SLE() :2 0;1)) to which these notes are devoted, corresponding to the parameter value = 0. Think of ( ) as progressively t t0 eating away the upper half-plane so that the subdomain H =HnK remains at time t, t t whereK = (0;t =f :s2 (0;tg. There is a conformal isomorphismg :H H given t s t t by p 2 g (z) = z + 4t t which has the following asymptotic behaviour asjzj1 2t 2 g (z) =z + +O(jzj ): t z In particular g (z)z 0 asjzj1. As we shall show in Proposition 4.3, there is only t one conformal isomorphism H H with this last property. Thus we can think of the t family of maps (g ) as a canonical encoding of the path ( ) . t t0 t t0  Consider the vector eld b onHnf0g de ned by 2 2(xiy) b(z) = = : 2 2 z x +y  Fix z2Hnf0g and de ne ( 2 y =4; if z =iy (z) = infft 0 : =zg = t 1; otherwise.  Then (z) 0, and z2K if and only if (z)6t. Set z =g (z). Then for t(z) t t t 2 p z _ = =b(z ) (2) t t 2 z + 4t t 7 and, if (z) 1, then z 0 as t (z). Thus (g (z) : z 2 Hnf0g; t (z)) is t t  the (unique) maximal ow of the vector eld b in Hnf0g. By maximal we mean that (z : t (z)) cannot be extended to a solution of the di erential equation on a longer t time interval. 1.5 Loewner evolutions Think of SLE(0) as obtained via the associated ow (g ) by iterating continuously a t t0 p map g , which nibbles an in nitesimal piece (0; 2i t ofH near 0. Charles Loewner, in t the 1920's, studied complex domains H =Hn (0;t for more general curves ( ) , by a t t t0 similar continuous iteration of conformal maps, obtained now by considering the ow of a  time-dependent vector eldH of the form 2 b(t;z) = ; t 0; z2H: z t Here, ( : t 0) is a given continuous real-valued function, which is called the driving t function or Loewner transform of the curve . We shall study this ow in detail below, showing that it always provides a construction of a family of domains (H : t 0), and t sometimes also a path . Note that the ow lines (g (z)) for SLE(0) separate, left t t0 and right, each side of the singularity at 0, with the path ( ) growing up between the t t0 left-moving ow lines and the right-moving ones. In the general case, assuming that the qualitative picture remains the same, when we move the singularity point to the left, we t may expect that some left-moving ow lines are de ected to the right, so the curve ( ) t t0 turns to the left. Moreover, the wilder the uctuations of ( ) , the more convoluted we t t0 may expect the resulting path ( ) to be. t t0 Oded Schramm, in 1999, realized that for some conjectured conformally invariant scal- ing limits ( ) of planar random processes, with a certain spatial Markov property, the t t0 process ( ) would have to be a Brownian motion, of some di usivity . The asso- t t0 ciated processes ( ) were at that time totally new and have since revolutionized our t t0 understanding of conformally invariant planar random processes. 82 Brownian motion and harmonic functions We rst prove a conformal invariance property of complex Brownian motion, due to L evy. Then we prove Kakutani's formula relating Brownian motion and harmonic functions, and deduce from this the maximum principle for harmonic functions and the maximum modulus principle for holomorphic functions. This used to prove the Schwarz lemma. 2.1 Conformal invariance of Brownian motion 0 0 Theorem 2.1. LetD andD be domains and let :DD be a conformal isomorphism. 0 0 Fixz2D and setz =(z). Let (B ) and (B ) be complex Brownian motions starting t t0 t0 t 0 from z and z respectively. Set 0 0 0 T = infft 0 :B 62Dg; T = infft 0 :B 62Dg: t t R T 0 2 Set T = j (B )j dt and de ne for tT t 0  Z  s 0 2 (t) = inf s 0 : j (B )j dr =t ; B =(B ): r t (t) 0 0 0 0 Then (T; (B ) ) and (T ; (B ) ) have the same distribution. t tT tT t Figure 2: A Brownian motion stopped upon leaving the unit square, and its image under a conformal transformation 1  Proof. Assume for now that D is bounded and  has a C extension to D. Then T 1 3 almost surely and we may de ne a continuous semimartingale Z and a continuous adapted 3 Here and below, where we use notions depending on a choice of ltration, such as martingale or stopping time, unless otherwise stated, these are to be understood with respect to the natural ltration (F ) of (B ) . t t0 t t0 94 process A by setting Z Tt 0 2 Z =(B ) + (B B ); A = j (B )j ds + (t (Tt)): t Tt t Tt t s 0 Moreover, almost surely, A is an (increasing) homeomorphism of 0;1), whose inverse is an extension of . Denote the inverse homeomorphism also by . Write  = u +iv, B =X +iY and Z =M +iN . By It o's formula, for tT , t t t t t t u u v v dM = (B )dX + (B )dY; dN = (B )dX + (B )dY t t t t t t t t t t x y x y and so, using the CauchyRiemann equations, 0 2 dMdM =j (B )j dt =dA =dNdN; dMdN = 0: t t t t t t t t On the other hand, for tT , dM =dX; dN =dY; dMdM =dt =dA =dNdN; dMdN = 0: t t t t t t t t t t t 2 2 Hence (M ) , (N ) , (M A ) , (N A ) and (MN ) are all continuous local t t0 t t0 t t0 t t0 t t t0 t t martingales. Set M = M and N = N . Then, by optional stopping, (M ) , s (s) s (s) s s0 2 2 (N ) , (M s) , (N s) and (M N ) are continuous local martingales for s s0 s0 s0 s s s0 s s the ltration (F ) , whereF =F . De ne (Z ) by Z = M +iN . Then, by s s0 s (s) s s0 s s s L evy's characterization of Brownian motion, (Z ) is a complex (F ) -Brownian motion s s0 s s0 0 starting fromz =(z). NowB =Z fortT and, since is a bijection,T = infft 0 : t t 0 Z 62Dg. So we have shown the claimed identity of distributions. t 1  In the cases where D is not bounded or  fails to have a C extension to D, choose a 0  sequence of bounded open sets D "D with D D for all n. Set D =(D ) and set n n n n 0 0 0 T = infft 0 :B 62Dg; T = infft 0 :B 62Dg: n t n n t n R T n 0 2 0 0 1 SetT = j (B )j dt. ThenT "T andT "T almost surely asn1. Since isC n t n n 0 0 0  onD , we know that (T ; (B ) ) and (T ; (B ) 0 ) have the same distribution for alln, n n t tT tT n t n n which implies the desired result on letting n1. Corollary 2.2. Let D be a proper simply connected domain. Fix z2 D and let (B ) t t0 be a complex Brownian motion starting from z. Set T (D) = infft 0 : B 62 Dg. Then t P (T (D)1) = 1. z Proof. There exists a conformal isomorphism  : D D. By conformal invariance of Brownian motion ((B ) : t T (D)) is a time-change of Brownian motion run up to the t nite time when it rst exits from D. Hencej(B )j 1=2 eventually as t" T (D). But t (B ) is neighbourhood recurrent so visits the open setfz2 D :j(z)j 1=2g at an t t0 unbounded set of times almost surely. Hence T (D)1 almost surely. 4 Whereas It o calculus localizes nicely with respect to stopping times, and we exploit this, L evy's characterization of Brownian motion is usually formulated globally. The extension of Z andA beyond the exit time T exploits the robustness of It o calculus to set up for an application of L evy's characterization without localization. 102.2 Kakutani's formula and the circle average property A real-valued functionu de ned on a domainDC is harmonic ifu is twice continuously di erentiable on D with   2 2 u = + u = 0 2 2 x y everywhere on D. Harmonic functions can often be recovered from their boundary values using Brownian motion. Theorem 2.3 (Kakutani's formula). Let u be a harmonic function de ned on a bounded  domain D and having a continuous extension to the closure D. Fix z2D and let (B ) t t0 be a complex Brownian motion starting from z. Set T (D) = infft 0 :B 62Dg. Then t u(z) =E (u(B )): z T (D) 2 Proof. Suppose for now that u is the restriction to D of a C function onC. Denote this function also by u. De ne (M ) by the It o integral t t0 Z t M =u(z) + ru(B )dB : t s s 0 Then (M ) is a continuous local martingale. By It o's formula, u(B ) =M for allt6T . t t0 t t T Hence the stopped process M is uniformly bounded and, by optional stopping, u(z) =M =E (M ) =E (u(B )): 0 z T z T (D) 2 For each n2 N, the restriction of u to D =fz2 D : dist(z;D) 1=ng has a C n extension to C, regardless of whether u itself does. The preceding argument then shows that u(z) = E (u(B )) for all z2 D . Now T (D )" T (D) 1 as n1 almost z T (D ) n n n  surely. Since B is continuous and u extends continuously to D, we obtain the desired identity by bounded convergence on letting n1. In fact, the validity of Kakutani's formula, even just in the special case where D is a disc centred at z, turns out to be a useful characterization of harmonic functions. We will use the following result in Section 3. A proof may be found in 2. Proposition 2.4. Let D be a domain and let u : D 0;1 be a measurable function. Suppose that u has the following circle average property: for all z 2 D and any r 2 (0;d(z;D)), we have Z 2 1 i u(z) = u(z +re )d: 2 0 Then, either u(z) =1 for all z2D, or u is harmonic. 112.3 Maximum principle Kakutani's formula implies immediately that a harmonic functionu on a bounded domain  D, which extends continuously to D, cannot exceed the supremum of its values on the boundaryD. Moreover, as we now show, a harmonic function cannot achieve a maximum value on its domain, unless it is constant. Theorem 2.5 (Maximum principle). Let u be a harmonic function de ned on a domain D. Suppose there exists a point z2 D such that u(w) 6 u(z) for all w2 D. Then u is constant. Proof. It will suce to consider the case where u has a nite supremum value m, say, on D. Consider the set D =fz2 D : u(z) = mg. Then D is relatively closed in D, since 0 0 u is continuous. On the other hand, if z2 D , then for " 0 suciently small, the disc 0 B(z;") of radius " and centre z is contained in D. So, by Kakutani's formula Z 2 1 i m =u(z) = u(z +"e )d: 2 0 Sinceu is continuous and bounded above bym, this implies thatw2D wheneverjwvj = 0 ". HenceD is open. SinceD is connected,D can only be non-empty if it is the whole of 0 0 D. By the CauchyRiemann equations, the real and imaginary parts of a holomorphic function are harmonic. Hence, if f is holomorphic on a bounded domain D and extends  continuously toD, thenf may be recovered from its boundary values, just as in Kakutani's formula: for all z2D f(z) =E (f(B )) z T (D) and we have the estimate jf(z)j6 supjf(w)j: w2D Then a small variation on the argument for the maximum principle leads to the following result. Theorem 2.6 (Maximum modulus principle). Letf be a holomorphic function de ned on a domain D. Suppose there exists a point z2D such thatjf(w)j6jf(z)j for all w2D. Then f is constant. We can now prove Lemma 1.2. Proof of the Schwarz lemma. Let f : D D be a holomorphic function with f(0) = 0. Consider the function g(z) = f(z)=z. By Taylor's theorem, g is analytic and hence holomorphic inD. Fix z2D and r2 (jzj; 1). Then 1 jg(z)j6 supjg(w)j6 : r jwj=r 12Letting r 1, we getjg(z)j 6 1 and hencejf(z)j 6jzj for all z2D. Ifjf(z)j =jzj for i some z6= 0, thenjg(z)j = 1, say g(z) = e . Then g is constant on D by the maximum i modulus principle, so f(w) =e w for all w2D. 133 Harmonic measure and the Green function 3.1 Harmonic measure Harmonic measures are objects of potential theory. Here, we will consider harmonic mea- sures in the particular context of planar domains, introducing them through their interpre- tation as the hitting distributions of Brownian motion. We further con ne our attention to the case of proper simply connected domains. Let D be such a domain and let D be its Martin boundary. Let (B ) be a complex Brownian motion starting from z2D and t t0 consider the rst exit time T = T (D) as in Section 2.2. We have shown that T 1  almost surely. In the case D = D and z = 0, we know that B converges in D as t" T , t with limitB uniformly distributed on the unit circle. In general, there exists a conformal T isomorphism :DD takingz to 0. Then, by conformal invariance of Brownian motion, as t"T , B converges in D to a limit B 2D. Denote by h (z;:) the distribution of B t T D T on D. We call h (z;:) the harmonic measure for D starting from z. By the argument D used for Kakutani's formula, if u is a harmonic function on D which extends continuously 5 to D, then we can recover u from its boundary values by Z u(z) =E (u(B )) = u(s)h (z;ds): z T D D We can compute h (z;:) as follows. By conformal invariance of Brownian motion, for D i i 1 2 s ;s 2D and  ; 2 0; 2) with  6 and (s ) =e and (s ) =e , we have 1 2 1 2 1 2 1 2   2 1 i i 1 2 h (z; s ;s ) =P (B 2 s ;s ) =P (B 2 e ;e ) = : D 1 2 z T 1 2 0 T (D) 2 We often x an interval IR and a parametrization s :ID of the Martin boundary. We may then be able to nd a density function h (z;:) on I such that D Z t 2 h (z;t)dt =h (z; s(t );s(t )) D D 1 2 t 1 i(t) 6 If we determine  as a continuous function on I such that e =(s(t)), then 1 d h (z;t) = : D 2dt The following two examples are not only for illustration but will also be used later. it Example 3.1. Take D = D and parametrize the boundary as (e : t2 0; 2)). Fix w =x +iy2D and recall from Section 1.3 the conformal automorphism  onD taking 0;w 5 This is not Kakutani's formula, unless D is a Jordan domain. For example, if D = Hn (0;i, then the requirement that u extend continuously to D imposes that u have a limit at1 but allows di erent boundary values on each side of the slit 0;i). 6 Note that the function  on I is determined uniquely by D and s up to an additive constant. 14i it it w to 0. The boundary parametrizations are then related by e = (e w)=(1w e ). On di erentiating with respect to t, we nd an expression for d=dt, and hence obtain 2 2 2 1 1jwj 1 1x y h (w;t) = = ; 06t 2: D it 2 2 2 2je wj 2 (costx) + (sinty) Example 3.2. Take D =H with the obvious parametrization of the boundary by R. Fix w =x +iy2H and consider the conformal isomorphism :HD takingw to 0 given by i (z) = (zw)=(zw ). The boundary parametrizations are related bye = (tw)=(tw ), so   1 1 y h (w;t) = Im = ; t2R: H 2 2  tw ((tx) +y ) 3.2 An estimate for harmonic functions (?) We will mark with (?) some sections and proofs which might be omitted on a rst reading. The following lemma allows us to bound the partial derivatives of a harmonic function in terms of its supremum norm. In conjunction with the CauchyRiemann equations, this will later allow us to deduce estimates on a holomorphic function starting from estimates on its real part. We present it here to illustrate how explicit calculations of harmonic measure can be used as a tool to obtain general estimates. Lemma 3.3. Let u be a harmonic function in D and let z2D. Then u 4kuk 1 (z) 6 : x  dist(z;D) Proof. It will suce to show that, for all " 0, the estimate holds with 4 replaced by 4(1 +"). Fix " 0. By scaling and translation, we reduce to the case where z = 0 and  dist(0;D) = 1 +". Then u is continuous onD so, for z2D, Z 2 i u(z) = u(e )h (z;)d: D 0 On di erentiating the formula for the harmonic density obtained in Example 3.1, we see thatrh (;) is bounded on a neighbourhood of 0, uniformly in , withrh (0;) = D D (cos; sin)=. Hence we may di erentiate under the integral sign to obtain Z 2 1 i ru(0) = u(e )(cos; sin)d:  0 Then Z 2 u kuk 4kuk 4(1 +")kuk 1 1 1 (0) 6 j cosjd = = : x    dist(0;D) 0 153.3 Dirichlet heat kernel and the Green function We give a probabilistic de nition of these two functions associated to a domainD and derive some of their properties. This will be used later in our discussion of the Gaussian free eld. It will be convenient to have the following regularity property for exit probabilities of the Brownian bridge. Proposition 3.4. De ne for t2 (0;1) and x;y2D  (t;x;y) =P (X 2D for all s2 0;t) D s where (X ) be a Brownian bridge from x to y in time t. Then  symmetric in its s 06s6t D second and third arguments and is jointly continuous in all three. Proof. (?) A simple scaling and translation allows us to vary the time and endpoints of 2 the Brownian bridge. Thus, from a single Brownian bridge (W ) inR from 0 to 0 in s 06s61 time 1, we can realise (X ) with explicit dependence on t;x and y by s 06s6t p X = (1 (s=t))x + (s=t)y + tW ; 06s6t: s s=t This makes clear that  (t;x;y) = (t;y;x), since (W ) is time-reversible. D D s 06s61 Fix "2 (0; 1=2) and de ne (") D(") =fz2D : dist(z;D)"g;  (t;x;y) =P (X 2D for all s2 "t; (1")t): s D Note that Z (") 0 0 (") 0 0 0 0  (t;x;y) =  ((1 2")t;x;y ) (x;y )dxdy D D 2 D (") where  is the joint density of (X ;X ). Then, for all sequences t t, x x, "t (1")t n n and y y, with obvious notation, we have n p p n jX X j6jx xj +jy yj +j t tj supjWj st n n n s st n 06s61 (" ) n (") 1 2 and   in L (D ), where " 2 (0; 1=2) is determined by (1 2" )t = (1 2")t n n n n for n suciently large. Hence lim inf (t ;x ;y ) (t;x;y) D n n n D(") n and (" ) (") n lim sup (t ;x ;y )6 lim (t ;x ;y ) = (t;x;y): D n n n n n n D D n n (") But  (t;x;y)  (t;x;y) and  (t;x;y)  (t;x;y) as " 0. Hence  is D(") D D D D continuous as claimed. 16De ne the Dirichlet heat kernel p on (0;1)DD and the Green function G on D D DD by Z 1 p (t;x;y) =p(t;x;y) (t;x;y); G (x;y) = p (t;x;y)dt (3) D D D D 0 2 1 jxyj =(2t) wherep(t;x;y) = (2t) e . Note that (t;x;x) 1 ast 0, soG (x;x) =1 D D for all x2D. The Green function is related directly to Brownian motion as an expected occupation density: thus, for allx2D and all non-negative measurable functionsf onD, we have Z Z T (D) G (x;y)f(y)dy =E f(B )dt: (4) D x t D 0 This follows from the de nition using Fubini's theorem and is left as an exercise. Extend G by 0 outside DD. Then, for any sequence of domains D " D, we D n have G (x;y)" G (x;y) for all x;y2 D. This follows from the de nition by monotone D D n convergence and is left as an exercise. We say that a domain D is Greenian if G (x;y)1 for some x;y2D. D Proposition 3.5. Every bounded domain is Greenian. Moreover, for any Greenian domain D, the Green function G is nite and continuous onf(x;y)2DD :x =6 yg. D  Proof. (?) For D bounded, there is a constant  0 such that  (1;x;y) 6 e for all D (t1) x;y2 D. Then, by the Markov property,  (t;x;y) 6 e for all t, so p (t;x;y) 6 D D 2  1 t  =(2t) e t e e wheneverjxyj . Then, by Proposition 3.4 and dominated conver- gence, G is nite and continuous away from the diagonal. In particular D is Greenian. D Fixy2D and choose a sequence of probability density functionsf , withf supported n n R infz2 D :jzyj 6 1=ng. Set g (x) = G (x;z)f (z)dz. Then g is a nite non- n D n n D negative measurable function on D and g G (:;y) as n1 locally uniformly on n D Dnfyg by continuity. Moreover, using the identity (4) and the strong Markov property, g has the circle average property onfz2 D :jzyj 1=ng. Hence G (:;y) has the n D circle average property on Dnfyg. Now take any domainD, xy2D and choose bounded domainsD "D. ThenG " n D n G so, by monotone convergence,G (:;y) has the circle average property onDnfyg. Hence D D G (:;y) is either identically in nite or harmonic on Dnfyg. Then, if D is Greenian, we D can use symmetry to see thatG is nite and continuous onf(x;y)2DD :x6=yg. D Conformal invariance of Brownian motion leads to a simple conformal invariance prop- erty for the Green function. Proposition 3.6. Let  : D (D) be a conformal isomorphism of planar domains. Then G ((x);(y)) =G (x;y); x;y2D: (D) D 17Proof. Since G "G for D "D, it will suce to consider the case where D and (D) D D n n are bounded when we know that G and G are nite and continuous away from the D (D) 0 2 diagonal. Fixx2D and a non-negative measurable functionf onD. Setg = (f)jj . Then, by the Jacobian formula, Z Z G ((x);(y))g(y)dy = G ((x);w)f(w)dw (D) (D) D (D) Z Z Z T ((D)) T (D) =E f(B )dt =E g(B )d = G (x;y)g(y)dy: (x) t x  D 0 0 D Hence G ((x);(y)) =G (x;y) for all y2D. (D) D For the upper half-plane, we can calculate explicity using the re ection principle p (t;x;y) =p(t;x;y)p(t;x;  y); x;y2H: H R b a=t b=t 1 x=t Then the integral (3) can be evaluated, using the formula e e = t e dx a and Fubini, to obtain 1 yx  G (x;y) = log : H  yx Then, by conformal invariance, every proper simply connected domain is Greenian. Also, by a suitable choice of, we get the following simple formula for the Green function of the unit disc logjyj G (0;y) = ; y2D: D  184 Compact H-hulls and their mapping-out functions A subset K of the upper half-plane H is called a compact H-hull if K is bounded and H =HnK is a simply connected domain. We shall associate to K a canonical conformal isomorphism g :HH, the mapping-out function of K. At the same time we associate K to K a real constant a , which we will identify later, in Section 6.2, as the half-plane K capacity of K. These are all basic objects of Loewner's theory, or more precisely of its chordal variant, where we consider evolution of hulls in a given domain towards a chosen boundary point. We shall see later that the theory has a property of conformal invariance which allows us to reduce the general case to the study of the special domain H with1 as the boundary point, which is mathematically most tractable. H=H\K K Figure 3: A compactH-hull. 4.1 Extension of conformal maps by re ection We start by explaining how a conformal isomorphism  :DH can be extended analyt- ically to suitably regular parts of the boundary D. We have already seen that  extends continuously to the Martin boundary but now we want more regularity. The idea is to re ect the domain across the boundary. Given a proper simply connected domain DH, de ne 0  0 D =fx2R :D is a neighbourhood of x inHg; D =DD fz  :z2Dg: 0 More generally, for any open set UD , de ne  D =DUfz  :z2Dg: U  As U varies, the sets D are exactly the open sets which are invariant under conjugation U   and whose intersection withH isD. Say that a functionf :D C is re ection-invariant U if    f (z ) =f (z); z2D : U 19Given a continuous function f on D, there is at most one continuous, re ection-invariant    function f on D extending f. Then f is the continuous extension by re ection of f. U  Such an extension f exists exactly when f has a continuous extension to DU which is real-valued on U. Any continuous extension by re ection of a holomorphic function is holomorphic, by an application of Morera's theorem. This is called the Schwarz re ection principle. Proposition 4.1. LetDH be a simply connected domain. LetI be a proper open subin- 0 terval ofR with ID and let x2I. Then there exists a unique conformal isomorphism  : D H which extends to a homeomorphism DI H (1; 1) taking x to 0. In particular I is naturally identi ed with an interval of the Martin boundary D. Moreover     extends further to a re ection-invariant conformal isomorphism  :D H . I (1;1)   Proof (?). Note thatD andH are proper simply connected domains. By the Riemann I (1;1)    mapping theorem, there exists a unique conformal isomorphism  : D H with I (1;1)   0    (z) =  (z ). Then  is  (x) = 0 and arg( ) (x) = 0. De ne  : D H by I (1;1) 0   a conformal isomorphism with (x) = 0 and arg (x) = 0. Hence  =  and so  is   1  re ection-invariant. Then  (I) (1; 1) and ( ) (1; 1)I, so (I) = (1; 1). Now   0  (D) is connected and does not meet (1; 1). Since arg( ) (x) = 0, by considering a   1 neighbourhood ofx, we must have (D)H. The same argument shows that ( ) (H)   D, so  (D) = H. Hence  restricts to a conformal isomorphism  : D H with the required properties.  On the other hand, any map with these properties has a continuous extension by   re ection toD , which is a bijection toH and is holomorphic by the Schwarz re ection I (1;1)   0    principle. Moreover (x) = 0, and arg( ) (x) = 0 since (I) = (1; 1). Hence = and so =. Proposition 4.2. Let D H be a simply connected domain and let  : D H be a conformal isomorphism. Suppose that  is bounded on bounded sets. Then  extends by   re ection to a conformal isomorphism  on D . 0 0 Proof (?). Fix x2 D and a bounded open interval I D containing x. Write  for x;I 1 the conformal isomorphism obtained in Proposition 4.1. Then f =  :HH is a x;I M obius transformation which is bounded, and hence continuous, on a neighbourhood of (1; 1) = (I) inH. Hence =f extends by re ection to a conformal isomorphism x;I x;I       =f  on D . The maps  must be consistent, and hence extend to a conformal I x;I I I    0 map  on D . Now  can only fail to be injective on D but, as a conformal map, can  only fail to be injective on an open set inC. Hence  is a conformal isomorphism. 4.2 Construction of the mapping-out function Given any compactH-hullK, we now specify a particular conformal isomorphismg =g : K HnK H. This will give us a convenient way to encode the geometry of K. We get uniqueness by requiring that g looks like the identity at1. K 20

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